Green’s Function Approach to Solve a Nonlinear Second Order Four Point Directional Boundary Value Pr

IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 3 Ver. IV (May - June 2017), PP 01-08 www.iosrjournals.org

Green’s Function Approach to Solve a Nonlinear Second Order Four Point Directional Boundary Value Problem Goteti V R L Sarma1, Awet Mebrahtu 2, Mebrahtom Sebhatu 3 Department of Mathematics, Eritrea Institute of Technology, Mai-nefhi, Eritrea.

Abstract: In this article a four point boundary value problem associated with a second order differential equation involving directional derivative boundary conditions is proposed. Then its solution is developed with the help of the Green’s function associated with the homogeneous equation. Using this idea and an Iteration method is proposed to solve the corresponding nonlinear problem.

Key words: Green’s function, Schauder fixed point theorem, Vitali’s convergence theorem.

I. Introduction Non local boundary value problems raise much attention because of its ability to accommodate more boundary points than their corresponding order of differential equations [5], [8]. Considerable studies were made by Bai and Fag [2], Gupta [4] and Web [9]. This research article is concerned with the existence and uniqueness of solutions for the second order four point boundary value problem with directional type boundary conditions (1.1) u " f (t , u)  0, t [a, b] ,

u (a)  k1u(1 ), where

u (b)  k2u(2 ); is a given function

(1.2) and k1 , k2  R

The Green’s function plays an important role in solving boundary value problems of differential equations. The exact expressions of the solutions for some linear ODEs boundary value problems can be expressed by the corresponding Green’s functions of the problems. The Green’s function method will be used to obtain an initial estimate for shooting method. The Greens function method for solving the boundary value problem is an effect tools in numerical experiments. Some BVPs for nonlinear integral equations the kernels of which are the Green’s functions of corresponding linear differential equations. The undetermined parametric method we use in this paper is a universal method, the Green’s functions of many boundary value problems for ODEs can be obtained by similar method.

In (2008), Zhao discussed the solutions and Green’s functions for non local linear second-order Three-point boundary value problems. subject to one of the following boundary value conditions: i. ii. iii. iv. where k was the given number and is a given point.

In (2013), Mohamed investigate the positive solutions to a singular second order boundary value problem with more generalized boundary conditions. He consider the Sturm-Liouville boundary value problem with the boundary conditions , where are all constants, is a positive parameter and is singular at .

Also the existence of positive solutions of singular boundary value problems of ordinary differential equations has been studied by many researchers such as Agarwal and Stanek established the existence criteria for positive solutions singular boundary value problems for nonlinear second order ordinary and delay differential equations using the Vitali’s convergence theorem. Gatical et al proved the existence of positive solution of the problem with the boundary conditions , using the iterative technique and fixed point theorem for cone for decreasing mappings.

Recently Goteti V.R.L. Sarma et al., studied the solvability of a four point boundary value problem with ordinary boundary conditions u  f (t )  0, t [a, b] satisfying the boundary conditions

u (a)  k1u (1 ), u (b)  k2u(2 ); where DOI: 10.9790/5728-1303040108

are real constants. www.iosrjournals.org

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Green’s Function Approach to Solve a Nonlinear Second Order Four Point Directional Boundary This article is organized as follows: In section 2 we construct the Green’s function to the homogeneous BVP corresponding to (1.1) satisfying (1.2) and then using this we proved the existence and uniqueness of the solution of the boundary value problem (1.1) satisfying the condition (1.2). In section 3 we present the iterative method of solution to the corresponding non linear boundary value problem. We illustrated our results by constructing a suitable example.

II. The Green’s function: We have the following conclusions: Theorem 2.1 Assume b  k2  a  k1. Then the Green’s function for the second-order four-point linear boundary value problem (1.1), (1.2) is given by

G (t , s )  K (t , s ) 

k1  b  k2  t  Kt (1 , s )  k2  k1  a  t  Kt (2 , s ) b  k2  a  k1

(2.1)

where

 ( s  a)(b  t ) ,ast b  ba  K (t , s)    (b  s)(t  a) ,at sb  ba

(2.2)

Proof: It is well known that the Green’s function is K(t, s) as in (2.2) for the second-order two-point linear boundary value problem

 u  f (t )  0, t  [a, b],  u (b)  0  u (a)  0,

(2.3)

And the solution of (2.3) is given by b

w(t )   K (t , s) f (s)ds,

(2.4)

a

and b

w(a)  0,

w(b)  0,

w( )   K ( , s) f ( s)ds.

(2.5)

a

The four-point boundary value problem (1.1), (1.2) can be obtained from replacing u(a) = 0, u(b) = 0 by u (a)  k1u(1 ) and u (b)  k2u(2 ) in (2.3). Thus, we suppose the solution of the four-point boundary value problem (1.1), (1.2) can be expressed by

u(t )  w(t )   c  dt   w(1 )  w(2 )

(2.6)

where c and d are constants that will be determined.

From (2.5), (2.6) we know that

u(a)   c  da   w(1 )  w(2 ) u(b)   c  db   w(1 )  w(2 )

u(1 )  w(1 )   c  d1   w(1 )  w(2 )

u(2 )  w(2 )   c  d2   w(1 )  w(2 ) Putting this into (1.2) yields

c  w(1 )  w(2 )  d  a  k1   w(1 )  w(2 )  k1w(1 )    c  w(1 )  w(2 )  d  b  k2   w(1 )  w(2 )  k2 w(2 ) Since b  k2  a  k1 , solving the system of linear equations on the unknown numbers c, d, using Cramer’s rule we obtain DOI: 10.9790/5728-1303040108

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Green’s Function Approach to Solve a Nonlinear Second Order Four Point Directional Boundary

  k1w(1 )  b  k2    k2 w(2 )  a  k1  c   b  k2  a  k1   w(1 )  w(2 )     k1w(1 )   k2 w(2 ) d   b  k2  a  k1   w(1 )  w(2 )  Hence, the solution of (1.1) with the boundary condition (1.2) is

u (t )  w(t )  (c  dt )  w(1 )  w(2 )   w(t ) 

k1 (b  k2  t ) w(1 )  k2 (k1  a  t ) w(2 ) b  k2  a  k1

This together with (2.4) implies that b

u(t )   K (t , s) f (s)ds  a

k1 (b  k2  t ) b  k2  a  k1

b

 Kt (1, s) f (s)ds  a

k2 (k1  a  t ) b  k2  a  k1

b

 Kt ( , s) f (s)ds 2

a

Consequently, the Green’s function G(t, s) for the boundary value problem (1.1), (1.2) is as described in (2.1).

From Theorem 2.1 we obtain the following corollary.

Corollary 2.1. If b  k2  a  k1 , then the second-order four-point linear boundary value problem

 u  f (t )  0, t  [a, b],  u (b)  k2u(2 )  u (a)  k1u(1 ), b

has a unique solution

u(t )   G(t , s) f (s)ds where G(t, s) is as in (2.1). a

Proof: Assume that the second-order four-point linear boundary value problem (2.7) has two solutions u(t) and v(t), that is

 u  f (t )  0, t  [a, b],  u (b)  k2u(2 )  u (a)  k1u(1 ),  v  f (t )  0, t  [a, b],  v(b)  k2v(2 )  v(a)  k1v(1 ), Let

(2.7)

and

(2.8)

z(t )  v(t )  u(t ), t a, b

(2.9)

Then (2.7) and (2.8)

z(t )  v(t )  u(t )  0, t a, b ,

therefore

z (t )  C1t  C2 , and z(t )  C1.

(2.10) where C1 and C 2 are undetermined constants. From (2.7), (2.8) and (2.9) we have z (a)  v(a)  u (a)  k1 z (1 ), (2.11) z (b)  v(b)  u (b)  k2 z (2 ).

(2.12)

Using (2.10) we obtain

z (a)  C1a  C2 ,

(2.13)

z (b)  C1b  C2 , z (1 )  C1 ,

(2.14) (2.15)

z(2 )  C1 , DOI: 10.9790/5728-1303040108

(2.16) www.iosrjournals.org

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Green’s Function Approach to Solve a Nonlinear Second Order Four Point Directional Boundary From (2.11), (2.13) and (2.15) we know that

C1  a  k1   C2  0,

(2.17)

and from (2.12), (2.14) and (2.16) we know that

C1 b  k2   C2  0,

(2.18)

Solving the system of equations (2.17) and (2.18), we get C1  0 and C2  0 .

Therefore

z(t )  0, t a, b , so u(t )  v(t ), t a, b , that is uniqueness of the solution.

Corollary 2.2. Suppose the nonlinear function g(t, u) is continuous on

a, b

, then if b  k2  a  k1 , the

nonlinear four-point boundary value problem

 u  g (t , u )  0, t  [a, b],  u (b)  k2u(2 )  u (a)  k1u(1 ), b

is equivalent to the nonlinear integral equation

u(t )   G(t , s) g  s, u(s)  ds where G(t, s) as in (2.1) a

If the endpoints of the interval are a  0, b  1 in the boundary condition, from Theorem 2.1, Corollaries 2.1 and 2.2 we obtain the following corollary.

Corollary 2.3. If 1  k2  k1 then the Green’s function for the second-order four-point linear boundary value problem

 u  f (t )  0, t  [0,1],  u (1)  k2u(2 )  u (0)  k1u(1 ), is

G (t , s )  B(t , s) 

(2.19)

k1 1  k2  t  Bt (1 , s )  k2  k1  t  Bt (2 , s ) 1  k2  k1

(2.20)

where

 s (1  t ), 0  s  t  1 B(t , s )   (1  s )t , 0  t  s  1

(2.21) 1

Hence the problem (2.19) has a unique solution If g(t, u) is continuous on

0,1 

u(t )   G(t , s) f  s  ds.

0

, then the nonlinear four-point boundary value problem

u  g (t , u )  0, t [0,1],  u (1)  k2u(2 ) u (0)  k1u(1 ), 1

is equivalent to the nonlinear integral equation

u(t )   G(t , s) g  s, u(s)  ds.

0

Example: Consider the second-order four-point boundary value problem:

u  cos t  0, t  [0,1],   1 1 1  1  u (1)  u   u (0)  u   , 2 6 3 5  Since 1  k 2

 1

1 4 1  and 0  k1  are not equal, from (2.20), the Green’s function is: 3 3 2

DOI: 10.9790/5728-1303040108

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Green’s Function Approach to Solve a Nonlinear Second Order Four Point Directional Boundary

G (t , s )  B(t , s) 

k1 1  k2  t  Bt (1 , s )  k2  k1  t  Bt (2 , s ) 1  k2  k1

1 1 , 2  3 5  4  3t  B  1 , s    2t  1 B  1 , s  G (t , s)  B(t , s)  where t t   11 11 6  5  1  t  s, 0  s  t  1 B(t , s)   (1  s)t , 0  t  s  1  s, 0  s  t  1 Bt (t , s )   which implies (1  s ), 0  t  s  1  s, 0  s  1  1   s , 0  s  2  1 and Bt (1 , s)   Bt (2 , s)   1  s, 0  1  s  1  1  s, 0  2  s  1 where k1 

1 , 2

1 6

1  ,

k2 

Hence the solution of second-order four-point boundary value problem is:

 1   1  u (t )   G (t , s) f  s  ds    B(t , s)   4  3t  B  , s    2t  1 B  , s   f  s  ds 6   5  0 0 1

1

 cos(t ) 

3(2t  1) 3t  4  1  2t  1  1  6t  8 sin    sin    cos(1)  11 11  6  11  5  11 III. Application to nonlinear problem:

In this section, we study the iterative solutions for the following nonlinear four-point boundary value problem

u  f (t , u )  0, t  (0,1),  u (1)  k2u(2 ) u (0)  k1u(1 ), (3.1)

0,1 , . k2  1  k1 J   0,1 , I  0,1 ,

with 1 ,2  Let





 0,  ,



D  x  C ( I ) M x  mx  0, such that mx 1  t   x(t )  M x 1  t  , t  I .

Concerning the function f we impose the following hypotheses:

 f (t , u ) is nonnegative continuous on J   ,    f (t , u ) is monotone increasing on u, for fixed t  J,  q   q  (0,1) such that f (t , ru )  r f (t , u ), 0  r  1, (t , u )  J 



.

(3.2) Obviously, from (3.2) we obtain

f (t , u )   q f (t , u ),   1,

(t , u)  J 



.

(3.3)

We can see that if 0   i  1, ai (t ) are nonnegative continuous on J, for i = 0,1,2,…,m, then m

f (t , u )   ai (t )ui satisfy the condition (3.2). i 1

DOI: 10.9790/5728-1303040108

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Green’s Function Approach to Solve a Nonlinear Second Order Four Point Directional Boundary Concerning the boundary value problem (3.1), we have following conclusions.

Theorem 3.1. Suppose the function f(t,u) satisfy the condition (3.2), it may be singular at t=0 and/or t=1, and 1

0   f (t ,1  t )dt  .

(3.4)

0

Then nonlinear singular boundary value problem (3.1) has a unique solution w(t) in C ( I )  C ( J ).

Constructing successively the sequence of functions 2

1

hn (t )   G(t , s) f ( s, hn1 ( s))ds,

n  1, 2,....

0

for any initial function h0 (t )  0(  0), t  I then of convergence is



(3.5)

hn (t ) must converge to w(t) uniformly on I and the rate



max hn (t )  w(t )  O 1  N q , tI

n

(3.6)

where 0< N <1 , which depends on the initial function h0 (t ), G (t , s ) as in (2.20).

Proof. Let





P  x(t ) x(t )  C  I  , x(t )  0 , 1

Fx(t )   G(t , s) f (s, x(s))ds, x(t )  D.

(3.7)

0

F : D  P is increasing; From Corollary 2.3 we know that if u  D satisfies (3.8) u(t )  Fu(t ), t  I , 1 2 then u  C  I   C  J  is a solution of (3.1).

It is easy that the operator

For any

x  D , there exist positive numbers 0  mx  1  M x such that mx (1  s)  x( s)  M x (1  s), s  I ,

 mx 

q

f (s,1  s)  f (s, x(s))   M x  f (s,1  s),

By (2.20) and (2.21) we have

G (t , s )  B(t , s ) 

q

G (t , s )  B(t , s) 

s  J.

(3.9)

k1 1  k2  t  Bt (1 , s )  k2  k1  t  Bt (2 , s ) 1  k2  k1

k1 (1  k2  t ) Bt (1 , s )  k2 ( k1  t ) Bt (2 , s) 1  k2  k1

 k2 (k1  t )  Bt (2 , s) 1  t    G (t , s )  (1  t ) 1  k2  k1 k B ( , s)  k2 (k1  t ) Bt (2 , s) G (t , s)  t (1  t )  1 t 1 1  k2  k1 k1 Bt (1 , s )  

   k2 (k1  t )   k1 Bt (1 , s)   1  t  Bt (2 , s)      G (t , s)  (1  t ) 1  1  k2  k1    

(3.10)

(3.11)

Using (3.7), (3.3) and (3.9)-(3.11) and the conditions (3.2), we obtain

DOI: 10.9790/5728-1303040108

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Green’s Function Approach to Solve a Nonlinear Second Order Four Point Directional Boundary 1

Fx(t )   G (t , s) f ( s, x( s))ds 0

1

  (1  t )

 k2 (k1  t )  Bt (2 , s) q  1  t   mx  f (s,1  s) ds 1  k2  k1

k1 Bt (1 , s)  

0

 (1  t )  mx 

q





 k2 (k1  t )  1 1  1  t   B ( , s) f ( s,1  s) ds  B ( , s) f ( s,1  s) ds  , t  I      t 1 0 t 2  1  k2  k1 0 

k1  

(3.12) 1

Fx(t )   G (t , s ) f ( s, x( s ))ds 0

   k2 (k1  t )   k1 Bt (1 , s )   1  t  Bt (2 , s)  q     M x  f ( s,1  s ) ds   (1  t ) 1  1  k2  k1   0  



1

 (1  t )  M x 



   k2    k1  k1k2  B(1 , s)  1  t  k2  k1k2  B(2 , s)      f ( s,1  s )  ds, t  I 0 1  1  k1 1  k22   1  k2  k11       1

q

(3.13) By (3.4), (3.12) and (3.13) we obtain

F : D  D. For any ho  D , we let





lho  sup l  0 lho (t )   Fho  (t ), t  I ,





Lho  inf L  0 Lho (t )   Fho  (t ), t  I ,  m  min 1, lho  u0 (t )  mh0 (t ),

 

1 1 q

 , 

v0 (t )  Mh0 (t ),

 M  max 1, Lho  un (t )  Fun 1 (t )

(3.14)

 

vn (t )  Fvn 1 (t ),

1 1 q

  

n  0,1, 2,...

(3.15) Since the operator F is increasing, from (3.2), (3.14) and (3.15) we know that

u0 (t )  u1 (t )    un (t )   vn (t )    v1 (t )  v0 (t ), t  I . m , from (3.2), (3.7) and (3.15), it can obtained by induction that For t0  M

un (t )   t0  vn (t ), qn

t  I , n  0,1, 2, 

From (3.16) and (3.17) we know that



(3.16)

(3.17)



0  un  p (t )  un (t )  vn (t )  un (t )  1   t0  Mh0 (t ) , qn

n, p

(3.18)

so that there exist function w(t )  D such that DOI: 10.9790/5728-1303040108

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Green’s Function Approach to Solve a Nonlinear Second Order Four Point Directional Boundary

un (t )  w(t ),

vn (t )  w(t ), (uniformly on I),

(3.19)

un (t )  w(t )  vn (t ),

t  I , n  0,1, 2, 

(3.20)

and From the operator F is increasing and (3.15) we have

un 1 (t )  Fun (t )  Fw(t )  Fvn (t )  vn 1 (t ),

n  0,1, 2,  This together with (3.19) and uniqueness of the limit imply that w(t ) satisfy (3.8), hence

w(t )  C1  I   C 2  J  is a solution of (3.1).

From (3.5) and (3.15) and the operator F is increasing, we obtain

un (t )  hn (t )  vn (t ),

t  I , n  0,1, 2, 

(3.21)

thus, from (3.18), (3.20) and (3.21) we know

hn (t )  w(t )  hn (t )  un (t )  un (t )  w(t )



 2 vn (t )  un (t )  1   t0 



so that max hn (t )  w(t )  1   t0  tI

qn

qn

 M max h (t) . tI

 M h (t) , 0

0

So that (3.6) holds.

From h0 (t ) which is arbitrary in D we know that w(t) is the unique solution of the Eq. (3.8) in D. Suppose w1 (t ) is a

C1  I   C 2  J  solution of boundary value problem (3.1). Let

z (t )  w1 (t )  Fw(t ), t  I .

Similar to the proof of (2.9) in section 2 we obtain w1 (t )  w(t ), hence (3.1) in

w(t ) is the unique solution of Eq.

C  I  C  J  .

1

Remark: If

2

f t, u  is continuous on I X R+ then it is quite evident that the condition (3.4) holds. Hence the

unique solution

w t   C 2  J  .

References

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

Al-Hayan, W. (2007). A domain Decomposition method with Green’s functions for solving Twelfth-order of boundary value problems. Applied Mathematical sciences, Vol. 9, 2015, no. 8, 353 – 368.

Bender, C. M. and S. A. Orzag (1999). Advanced mathematical methods for scientists and Engineers; Asymptotic methods and perturbation theory, ACM30020.

Benchohra, M. et al Second-Order boundary value problem with integral boundary conditions. Boundary value problems article ID 260309 vol. 2011 Dr. Raisinghania, M. D. (2013), Integral equations and boundary value problems sixth edition; S. Chand & Company PVT. LTD. Greengard, L. and V. Kokhlin (1991). On the numerical solution of two-point boundary value problems. Communications on pure and applied mathematics vol. XLIV, 419-452(1991) Goteti V R L Sarma, Mebrahtom Sebhatu and Awet Mebrahtu, Solvability Of Four Point Nonlinear Boundary Value Problem – Intl Journal of Engineering Research and Applications. Volume 7, Issue 2 ( Part 4) February 2017 Pp 10 – 18.

Herron, I. H. Solving singular boundary value problems for ordinary differential equations. Caribb. J. Math. Comput. Sci. 15, 2013, 1- 30.

Kumlin, P. (2003/2004), A note on ordinary differential equations; TMA401/MAN 670 Functional Analysis. Mathematics Chalmers & GU Liu, Z., Kang, S. M and J. S. Ume (2009). Triple positive solutions of nonlinear third order boundary value problems. Taiwanese Journal of Mathematics Vol. 13, no. 3 pp955-971.

Mohamed, M. & W. A. W. Azmi (2013), positive solutions to solutions to a singular second order boundary value problems. Int. Journal of math. Analysis, Vol.7, 2013, no. 41, 2005-2017.

Raisinghania, M. D. (2011), Integral equations and boundary value problems sixth edition; S. Chand & Company PVT. LTD. New Delthi-110 055.

Teterina, A. O. (2013), The Green’s function method for solutions of fourth order nonlinear boundary value problem. The university of Tennessee, Knoxville Yang, C. & P. Wang (2007). Green’s function and positive solutions for boundary value problems of third order differential equations. Computers and mathematics with applications 54(2007)567-578.

Zhao, Z. (2007) positive solutions for singular three-point boundary value problems. Electronic Journal of Differential equations Vol. 2007(2007), no. 156, pp. 1-8. ISSN 1072-6691 Zhao, Z. (2007), Solution and green’s functions for linear second order three-point boundary value problems; Computers and mathematics with applications 56(2008)104-113.

DOI: 10.9790/5728-1303040108

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